{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/50697"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/50697","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Sturm-Liouville estimates for the spectrum and Cheeger constant","abstract":"Buser’s inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser’s inequality. Agol’s result is less transparent since it is given implicitly by a set of equations, one of which is a differential equation Agol could not solve except when M is three-dimensional. We show that a substitution transforms Agol’s differential equation into the Riemann differential equation. Then, we give a proof of Agol’s result and also generalize it using Sturm-Liouville theory. Under the same assumptions on M, we are able to give upper bounds on the higher eigenvalues of M , λ_k(M), in terms of the eigenvalues of a Sturm-Liouville problem which depends on h(M). We then compare the Weyl asymptotic of λ_k(M) given by the works of Cheng, Gromov, and Berard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given by Atkinson-Mingarelli.","abstract_html":"Buser’s inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser’s inequality. Agol’s result is less transparent since it is given implicitly by a set of equations, one of which is a differential equation Agol could not solve except when M is three-dimensional. We show that a substitution transforms Agol’s differential equation into the Riemann differential equation. Then, we give a proof of Agol’s result and also generalize it using Sturm-Liouville theory. Under the same assumptions on M, we are able to give upper bounds on the higher eigenvalues of M , λ_k(M), in terms of the eigenvalues of a Sturm-Liouville problem which depends on h(M). We then compare the Weyl asymptotic of λ_k(M) given by the works of Cheng, Gromov, and Berard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given by Atkinson-Mingarelli.","abstract_has_math":false,"creators":["Benson, Brian"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dunfield, Nathan M.","Laugesen, Richard S.","Alexander, Stephanie B.","Leininger, Christopher J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-09-16T17:25:22Z","date_published":"2014-09-16T17:25:22Z","updated_at":"2026-07-22T22:25:41Z","subjects":["Cheeger constant","spectrum of Laplacian","eigenvalues of closed Riemannian manifolds","Buser's inequality"],"languages":["en"],"rights":["Copyright 2014 Brian Benson"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/50697","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dunfield, Nathan M.","Laugesen, Richard S.","Alexander, Stephanie B.","Leininger, Christopher J."]},{"key":"dc:creator","label":"Author","values":["Benson, Brian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-09-16T17:25:22Z","2014-08","2014-09-16"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Cheeger constant","spectrum of Laplacian","eigenvalues of closed Riemannian manifolds","Buser's inequality"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Brian Benson"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/50697"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Buser’s inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). 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